By Jfry KSmit

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Example: To evaluate the eﬀectiveness of a medical testing procedure such as for disease screening or illegal drug use, we will evaluate the probability of a false negative or a false positive using the following notation T + : The test is positive T − : The test is negative D+ : The person has the disease D− : The person does not have the disease. The “sensitivity” of a test is the probability of a positive result given the person has the disease. 99. 99. 000001, ﬁnd the probability of a false positive P [D− |T + ].

As previously mentioned, the use of the Conjugate prior distribution has the extra advantage that the resulting posterior distribution is in the same family. Example: The prior distribution for the probability of heads when ﬂipping a certain coin is p(̺) ∝ ̺α−1 (1 − ̺)β−1 . 4) and the likelihood for a random sample subsequently taken is p(x|̺) ∝ ̺x (1 − ̺)n0 −x . 5) When these are combined to form the posterior distribution of ̺, the result is p(̺|x) ∝ p(̺)p(x|̺) ∝ ̺(α+x)−1 (1 − ̺)(β+n0 −x)−1 .

As the number of degrees of freedom increases, a random variate which follows the Scalar Student t-distribution t ∼ t(ν, t0 , σ 2 , φ2 ) approaches a Normal distribution t ∼ N (t0 , φ2 σ 2 ) [17, 41]. 7 F-Distribution The F-distribution [1, 22, 66] is used to describe continuous random variables which are strictly positive. 68) and transforming variables to x= x1 /ν1 . 69) In the derivation, x1 and x2 could be independent sums or squared deviations of standard Normal variates. 71) with x ∈ R+ , ν1 ∈ N ν2 ∈ N.

### Applied Statistics and the SAS Programming Language by Jfry KSmit

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